Spinning single component around axis without affecting the compounding units
Are you trying to make an animation or just a still image? If you want a still image, I've had a sort of similar situation where I resorted to a simple cheat. I needed to have a guitar string reach up to a tuning peg, and then wrap around it repeatedly. 

My solution was to make 2 strings. One string reaches up to where the curve is supposed to begin, then cuts off. Where that string ended, the other string starts. 

The other string is made by just creating a single flat circle, positioned right at the end of the first string, whose size and number of vertices was the same as a cross section of the first string. 

Its face is perpendicular to the top of the peg, and I positioned it close to the peg. Then I added the screw modifier, chose the peg as the axis object, and messed around with the iterations, steps, and angle settings to create a coil. 

The modifier can be a little confusing. If it doesn't seem to wrap exactly around the central rod of the yoyo, make sure that rod has its scale and rotation applied in object mode (ctrl+A). Then make sure the origin of the rode is in the middle of it (ctrl+alt+shift+C). For the string itself, you can reposition the origin (same command) to different spots to control how tight the coil is. If you move the small disc that creates the string up and down in edit mode (well, in my case it's up and down) then the coil gets more compressed. If you tab into edit mode and move it, the entire coil just moves vertically without getting tighter or looser. So to get the modifier to cooperate, you have to pay attention to the position of the origin of everything, and whether you're moving things in edit more or object mode.